Lemma 10 (Lemma 19, [1]})
For \(\xi > 0\) , define \(I_{\xi } \triangleq [\mu _1-\xi , \mu _1+\xi ]\times \ldots [\mu _K-\xi , \mu _K+\xi ]\) . And for \(T\ge 1\) , consider the event: \(\mathcal {E}_T = \bigcap \limits _{t = {T^{1/4}}}^T \big (\widehat{{\mu }}_t \in I_{\xi } \big )\) . Then there exists two constants \(B, C\)  that only depend on \({\mu }\)  and \(\xi \)  such that
\(\mathbb {P}\big (\mathcal {E}_T^c\big ) \le B T \exp (-C T^{1/8})\) 
